Problem: Solve for $x$ and $y$ using elimination. $\begin{align*}-2x+y &= -4 \\ -5x+y &= -8\end{align*}$
We can eliminate $y$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $1$ $\begin{align*}2x-y &= 4\\ -5x+y &= -8\end{align*}$ Add the top and bottom equations. $-3x = -4$ Divide both sides by $-3$ and reduce as necessary. $x = \dfrac{4}{3}$ Substitute $\dfrac{4}{3}$ for $x$ in the top equation. $-2( \dfrac{4}{3})+y = -4$ $-\dfrac{8}{3}+y = -4$ $y = -\dfrac{4}{3}$ $y = -\dfrac{4}{3}$ The solution is $\enspace x = \dfrac{4}{3}, \enspace y = -\dfrac{4}{3}$.